The Method of Direct Integration Examples 2

Recall from The Method of Direct Integration page that if we have a differential equation in the form $\frac{dy}{dt} = ay + b$ where $a$ and $b$ are constants, then the general solution can be obtained by rewriting this differential equation as to get the lefthand side to be the derivative of a natural logarithm while in general, the solution is given as:

(1)

\begin{align} y = De^{at} - \frac{b}{a} \end{align}

Let's look at some examples of solving differential equations by direct integration.

Example 1

Find all solutions to the differential equation $\frac{dy}{dt} = 11y + 132$.

We will first rewrite out differential equation and solve as follows:

(2)

\begin{align} \quad \frac{dy}{dt} = 11(y + 12) \\ \quad \frac{\frac{dy}{dt}}{y + 12} = 11 \\ \quad \frac{d}{dt} \ln \mid y + 12 \mid = 11 \\ \quad \int \frac{d}{dt} \ln \mid y + 12 \mid \: dt = \int 11 \: dt \\ \quad \ln \mid y + 12 \mid = 11t + C \\ \quad \mid y + 12 \mid = e^{11t + C} \\ \quad y + 12 \mid = \pm e^{11t}e^C \\ \quad y = De^{11t} -12 \end{align}

Note that if $D = 0$ then we get $y = 12$ which is a solution to our differential equation, and so all solutions are given by $y = De^{11t} - 12$ for $D$ as a constant.

Example 2

Find all solutions to the differental equation $y \frac{dy}{dt} - 4y^2 + 36y = 0$.

If we rewrite the differential equation above as $\frac{dy}{dt} = 4y - 36$ by dividing all terms by $y$ (provided $y \neq 0$) and then rearranging terms. Thus we solve this differential equation as:

(3)

\begin{align} \quad \frac{dy}{dt} = 4y - 36 \\ \quad \frac{dy}{dt} = 4(y - 9) \\ \quad \frac{\frac{dy}{dt}}{y - 9} = 4 \\ \quad \frac{d}{dt} \ln \mid y - 9 \mid = 4 \\ \quad \int \frac{d}{dt} \ln \mid y - 9 \mid \: dt = \int 4 \: dt \\ \quad \ln \mid y - 9 \mid = 4t + C \\ \quad \mid y - 9 \mid = e^{4t + C} \\ \quad y - 9 = \pm e^{4t}e^C \\ \quad y = D e^{4t} + 9 \end{align}

Note that if $D = 0$ we get that $y = 9$ which is a solution to our differential equation. Also, $y = 0$ (which we initially omitted) is also a solution to our differential equation. Thus all solutions are given by $y = De^{4t} + 9$ for $D$ as a constant, and $y = 0$.